JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

The mean and variance of 5 observations are 5 and 8 respectively. If 3 observations are 1, 3, 5 then the sum of cubes of the remaining two observations is:

Choose the correct answer:

Explanation

Solution

1. Use the Mean to find the sum of remaining observations:

Let the five observations be $x_1, x_2, x_3, x_4, x_5$ . Given $x_1=1, x_2=3, x_3=5$ . Let the remaining two be $a$ and $b$ .

\text{Mean} = \frac{1+3+5+a+b}{5} = 5

9 + a + b = 25 \implies a + b = 16 \quad \dots \text{(Eq. 1)}

2. Use the Variance to find the sum of squares:

\text{Variance} (\sigma^2) = \frac{\sum x_i^2}{n} - (\text{Mean})^2 = 8

\frac{1^2 + 3^2 + 5^2 + a^2 + b^2}{5} - 5^2 = 8

\frac{1 + 9 + 25 + a^2 + b^2}{5} - 25 = 8

\frac{35 + a^2 + b^2}{5} = 33 \implies 35 + a^2 + b^2 = 165

a^2 + b^2 = 130 \quad \dots \text{(Eq. 2)}

3. Find the product $ab$ :

Using $(a+b)^2 = a^2 + b^2 + 2ab$ :

(16)^2 = 130 + 2ab \implies 256 = 130 + 2ab

2ab = 126 \implies ab = 63

4. Calculate the sum of cubes:

Using the identity $a^3 + b^3 = (a + b)(a^2 + b^2 - ab)$ :

a^3 + b^3 = (16)(130 - 63)

a^3 + b^3 = 16 \times 67 = 1072

Correct Option: (2)